Question

The ideas underlying Hamiltonian systems extend to higher-dimensional systems. For example, consider the threedimensional autonomous system $$ \begin{aligned} &x^{\prime}=f(x, y, z) \\ &y^{\prime}=g(x, y, z) \\ &z^{\prime}=h(x, y, z) \end{aligned} $$ (a) Use the chain rule to show that autonomous system (17) has a conserved quantity if there exists a function \(H(x, y, z)\) for which $$ \frac{\partial}{\partial x} H(x, y, z) f(x, y, z)+\frac{\partial}{\partial y} H(x, y, z) g(x, y, z)+\frac{\partial}{\partial z} H(x, y, z) h(x, y, z)=0 . $$ (b) Show that \(H(x, y, z)=\cos ^{2}(x)+y e^{z}\) is a conserved quantity for the system $$ \begin{aligned} &x^{\prime}=y e^{z} \\ &y^{\prime}=y \cos x \sin x \\ &z^{\prime}=\cos x \sin x \end{aligned} $$

Short answer

Question: Show that the function H(x, y, z) has a conserved quantity for the given autonomous system and demonstrate the conditions required for this conservation. Answer: The function H(x, y, z) = cos²(x) + ye^z is a conserved quantity for the given autonomous system because the condition $$ \frac{\partial H}{\partial x} f(x, y, z) + \frac{\partial H}{\partial y} g(x, y, z) + \frac{\partial H}{\partial z} h(x, y, z) = 0 $$ is satisfied. This condition arises from the chain rule application on the total derivative of H with respect to time t, and ensures that H's total derivative with respect to time is zero.

Step-by-step solution

Apply the Chain Rule

To find the total derivative of H with respect to time t, we need to apply the chain rule. $$ \frac{d H}{dt} = \frac{\partial H}{\partial x} \frac{dx}{dt} + \frac{\partial H}{\partial y} \frac{dy}{dt} + \frac{\partial H}{\partial z} \frac{dz}{dt} $$ Now, substitute the functions f(x, y, z), g(x, y, z), and h(x, y, z) given in the autonomous system. $$ \frac{d H}{dt} = \frac{\partial H}{\partial x} f(x, y, z) + \frac{\partial H}{\partial y} g(x, y, z) + \frac{\partial H}{\partial z} h(x, y, z) $$ For the total derivative of H to be 0, we need the above equation to be equal to 0: $$ \frac{\partial H}{\partial x} f(x, y, z) + \frac{\partial H}{\partial y} g(x, y, z) + \frac{\partial H}{\partial z} h(x, y, z) = 0 $$ This condition, if true, will indicate that H is conserved in the system, since its total derivative with respect to time will be zero.

Verify the Conserved Quantity for the Given System

Now, we will verify that H(x, y, z) = cos²(x) + ye^z is, indeed, a conserved quantity for the provided autonomous system. We will find the partial derivatives of H with respect to x, y, and z and apply the condition from step 1: First, find the partial derivatives of H: $$ \begin{aligned} \frac{\partial H}{\partial x} &= -2\cos(x)\sin(x) \\ \frac{\partial H}{\partial y} &= e^z \\ \frac{\partial H}{\partial z} &= y e^z \end{aligned} $$ Now, substitute these derivatives and the functions f, g, h in the equation from step 1: $$ \begin{aligned} -2\cos(x)\sin(x) \cdot y e^{z} + e^z \cdot y \cos x \sin x + y e^z \cdot \cos x \sin x &= 0 \\ (-2\cos(x)\sin(x) y + y\cos(x)\sin(x) + y\cos(x)\sin(x))e^{z} &= 0 \end{aligned} $$ Since e^z is always positive, we can ignore it and focus on the multiplier of e^z: $$ -2\cos(x)\sin(x) y + y\cos(x)\sin(x) + y\cos(x)\sin(x) = 0 $$ This simplifies to: $$ y\cos(x)\sin(x) = 0 $$ Therefore, our condition from step 1 is satisfied, and H(x, y, z) = cos²(x) + ye^z is a conserved quantity for the given autonomous system.

Key concepts

Autonomous Differential Systems

An autonomous differential system is defined by a set of differential equations that do not explicitly depend on the independent variable, often time (t). In the context of a three-dimensional system like the one described in the exercise, the system's evolution is specified by the functions f(x, y, z), g(x, y, z), and h(x, y, z). These functions dictate how the variables x, y, and z change over time.

One powerful aspect of autonomous systems is their capacity for self-regulation, meaning the system's state at any given moment is determined solely by its current state, without reference to an external clock. The mathematical study of such systems provides insight into an array of natural phenomena, from the orbits of celestial bodies to the behavior of ecosystems.
For students tackling these equations, it's crucial to focus on understanding how these systems evolve over time by examining their dynamic behavior and identifying any conserved quantities that signify the system's invariance under certain conditions.

Conserved Quantities

In physics and mathematics, a conserved quantity of a system is a property that remains constant as the system evolves. This concept is closely related to symmetries and invariance and plays a significant role in the analysis and understanding of systems. For instance, in the context of the Hamiltonian systems mentioned, if we can find a function H(x, y, z) such that the expression provided in the exercise equals zero, then this function represents a conserved quantity.

A real-world application of conserved quantities is the conservation of energy in physics, which states that the total energy of an isolated system will remain constant over time. Detecting these quantities in a set of differential equations can drastically simplify the study of the system by reducing the number of variables to consider. Students should note that finding a conserved quantity often leads to deeper insights into the system's behavior and can simplify the process of solving the equations.

Chain Rule

The chain rule is a fundamental theorem in calculus used to compute the derivative of a composite function. When dealing with higher-dimensional autonomous systems, the chain rule becomes invaluable for determining how a function like H(x, y, z) changes with time, especially when x, y, and z themselves change with time.

As illustrated in the step-by-step solution, by applying the chain rule, students can unpack the total derivative in terms of partial derivatives and the rates of change of the variables x, y, and z. Here's a practical analogy to help comprehend this abstract concept: envision a hiker on a mountain terrain - the slope (derivative) of her path (composite function) at any point depends on both the slope of the mountain (the underlying functions) and the direction she is heading (the rates of change).

Partial Derivatives

Partial derivatives represent the rate at which a function changes as one of its variables is varied, while the others are held constant. Think of a partial derivative as the slope of the terrain in one direction. In multi-variable calculus, partial derivatives are essential for understanding how multi-dimensional surfaces behave.

In the solution provided for the problem, partial derivatives of H(x, y, z) are computed with respect to each variable, reflecting how H changes when only x, y, or z is varied independently. This step is pivotal in evaluating whether H is conserved. It's recommended for students to master partial differentiation as it's pivotal in fields like thermodynamics, quantum mechanics, and economics, where modeling the effect of changing certain parameters while keeping others constant is frequently required.